Problem: The $n\text{th}$ partial sum of the series $\sum\limits_{n=1}^{\infty }{{{a}_{n}}}$ is given by ${{S}_{n}}=\frac{n+5}{n-3}$. $a_{10}=$
Solution: ${{a}_{10}}={{S}_{10}}-{{S}_{9}}$ ${{a}_{10}}=\frac{15}{7}-\frac{14}{6}=-\frac{8}{42}=-\frac{4}{21}$